Manuscript X-F-ST@1 · X-F-ST.tex · section · e71f03ef

The retained action and the invariant-jet boundary

The retained action and the invariant-jet boundary · lines 598–660

X-F-ST@1#eq:Boperator, X-F-ST@1#eq:Euler

Here $G_{2}$ is the symmetric polarization of the coefficient in (eq:Eexp), not the second coefficient of a differently indexed Einstein tensor. Explicitly,

G_{2} [l, l] = \frac{1}{2} \frac{d ^{2}}{d s ^{2}}_{s = 0} (J_{η + s l} G^{μν} [η + s l]) .

It is an unambiguous constant-coefficient quadratic differential polynomial of total derivative order two. For a finite component basis it can be written

G_{2}^{I} [r_{1}, r_{2}] (y) = J, L, α, β ∣ α ∣ + ∣ β ∣ \leq 2 \sum b^{I, α β}_{J L} (\partial^{α} r_{1}^{J}) (y) (\partial^{β} r_{2}^{L}) (y) .

The constants are fixed by the displayed derivative definition; no independent cubic vertex is selected.

For completeness,

G_{lin, μν} [l] = \frac{1}{2} (\partial_{ρ} \partial_{μ} l^{ρ}_{ν} + \partial_{ρ} \partial_{ν} l^{ρ}_{μ} - □ l_{μν} - \partial_{μ} \partial_{ν} tr_{η} l - η_{μν} (\partial_{α} \partial_{β} l^{α β} - □ tr_{η} l)) .

Define

T^{μν} [q, ϕ] = q^{μα} q^{ν β} \partial_{α} ϕ \partial_{β} ϕ - \frac{1}{2} q^{μν} (q^{α β} \partial_{α} ϕ \partial_{β} ϕ - m^{2} ϕ^{2}), T_{0} = T [η, ϕ] .

Then

E_{q} E_{ϕ} V_{1, g} [l, ϕ] = \frac{1}{2 κ ^{2}} (E [q] - κ^{2} g J_{q} T [q, ϕ]), = - \partial_{μ} ([(1 - g) η^{μν} + g J_{q} q^{μν}] \partial_{ν} ϕ) - m^{2} (1 - g + g J_{q}) ϕ, = - \frac{1}{2} \int g l_{μν} T_{0}^{μν} [ϕ] d^{4} x, \frac{δ V _{1, g}}{δ l} = - \frac{1}{2} g T_{0} [ϕ] .

Displayed formulas in this passage

X-F-ST@1#eq:Boperator · line 612
$G_{2}^{I} [r_{1}, r_{2}] (y) = J, L, α, β ∣ α ∣ + ∣ β ∣ \leq 2 \sum b^{I, α β}_{J L} (\partial^{α} r_{1}^{J}) (y) (\partial^{β} r_{2}^{L}) (y) .$
X-F-ST@1#eq:Euler · line 645
$E_{q} E_{ϕ} V_{1, g} [l, ϕ] = \frac{1}{2 κ ^{2}} (E [q] - κ^{2} g J_{q} T [q, ϕ]), = - \partial_{μ} ([(1 - g) η^{μν} + g J_{q} q^{μν}] \partial_{ν} ϕ) - m^{2} (1 - g + g J_{q}) ϕ, = - \frac{1}{2} \int g l_{μν} T_{0}^{μν} [ϕ] d^{4} x, \frac{δ V _{1, g}}{δ l} = - \frac{1}{2} g T_{0} [ϕ] .$
unlabeled · line 604
$G_{2} [l, l] = \frac{1}{2} \frac{d ^{2}}{d s ^{2}}_{s = 0} (J_{η + s l} G^{μν} [η + s l]) .$
unlabeled · line 625
$G_{lin, μν} [l] = \frac{1}{2} (\partial_{ρ} \partial_{μ} l^{ρ}_{ν} + \partial_{ρ} \partial_{ν} l^{ρ}_{μ} - □ l_{μν} - \partial_{μ} \partial_{ν} tr_{η} l - η_{μν} (\partial_{α} \partial_{β} l^{α β} - □ tr_{η} l)) .$
unlabeled · line 636
$T^{μν} [q, ϕ] = q^{μα} q^{ν β} \partial_{α} ϕ \partial_{β} ϕ - \frac{1}{2} q^{μν} (q^{α β} \partial_{α} ϕ \partial_{β} ϕ - m^{2} ϕ^{2}), T_{0} = T [η, ϕ] .$

The retained action and the invariant-jet boundary

Displayed formulas in this passage

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